From Theorem 13.1 in [1]}, we are ready to state the condition in the following theorem.
[Fail-Free MAMPC]
If there exists some nonnegative function \(\gamma : \mathbb {X}_{0} \rightarrow \mathbb {R}_{+}\)  such that
\(\begin{aligned}(\mathbf {x}, u_{\textnormal {NN}}(\mathbf {x})) \in \mathbb {A}, \quad &\forall \mathbf {x} \in \mathbb {X}_{0}, \\L(\mathbf {x}) \le J^{*}(\mathbf {x}) + \gamma (\mathbf {x}), \quad &\forall \mathbf {x} \in \mathbb {X}_{0}, \\\gamma (\mathbf {x}) - c(\mathbf {x}, \mathbf {0}) < 0, \quad &\forall \mathbf {x} \in \mathbb {X}_{0} \setminus \mathbf {0},\end{aligned}\) 
